THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THERMAL STRESSES IN THICK-WALLED TUBES WITH LAMINAR CONVECTION HEAT TRANSFER Daniel P. Werner A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan April, 1968 IP-818

ACKNOWLEDGMENTS I am sincerely grateful to Professor V. S. Arpaci, Chairman of the doctoral committee, for his patience, invaluable suggestions, and personal interest during this investigation. I also wish to thank the remaining members of my doctoral committee for their advice and suggestions. The financial support of the experimental work by the Shell Oil Company is gratefully acknowledged, as is the financial support provided by the Department of Mechanical Engineering and the Horace H. Rackham School of Graduate Studies, University of Michigan. Finally, I wish to thank my wife for her patience, encouragement, and sacrifices. ii

TABLE OF CONTEENTS Page ACKNOWLEDGMENTS................................................ ii LIST OF FIGURES................................................ iv NOMENCLATURE................................................... vi CHAPTER I INTRODUCTION......................................... 1 II THEORETICAL ANALYSIS.............................. 4 A. Formulation........................... 4 B. Solution........................................ 13 1. Temperature Solution......................... 13 2. Stress Solution.............................. 20 C. Results.......................................... 24 III EXPERIMENTAL INVESTIGATION.........................o 41 IV DISCUSSION OF RESULTS............................... 48 APPENDICES A FORMULATION OF TEMPERATURE PROBLEM............... 49 B APPROXIMATE TEMPERATURE FORMULATION FOR THE ENTRANCE REGION.............................. 53 C EFFECT OF AXIAL CONDUCTION IN THE TUBE WALL........o 56 D TABULATED VALUES FOR MODIFIED GRAETZ SOLUTION....... 62 E THERMAL STRESS SOLUTION FOR PLANE STRAIN............ 69 BIBLIOGRAPHY.............................................. 71 iii

LIST OF FIGURES Figure 1 Schematic of System................................ 2 Effect of Modified Nusselt Number on Wall Temperature I. (R = 1)........................... 3 Effect of Modified Nusselt Number on Wall Temperature II. (R = 1, Pe = 100) 4 Effect of Modified Biot Number on Wall Temperature (R = Ro, Ro = 3, N = 20)............ 5 Effect of Tube Geometry on Wall Temperature (R = Ro, B = 3, N = 20).............................. 6 Comparison of Tangential and Axial Stress I. (R = 1, Ro = 3, B = 3, Pe = 10, N = 2)................ 7 Comparison of Tangential and Axial Stress II (R = Ro, Ro = 3 B = 3, Pe = 10, N-= 2).............. 8 Effect of Tube Geometry on Axial Stress I (R = = 3, Pe = 10, N = 2)....................... 9 Effect of Tube Geometry on Axial Stress II (R = Ro, B = 3, Pe = 10, N = 2)....................... 10 Effect of Modified Nusselt Number on Axial Stress (R = Ro, Ro = 1.6, B = 3, Pe = 10)................. 11 Effect of Modified Biot Number on Axial Stress (R = Ro, Ro = 1.6, Pe = 10, N = ).................... 12 Effect of Peclet Number on Axial Stress (R = Ro, Ro = 1.6, B 3 N = 6).................... 13 Effect of Modified Nusselt Number on Axial and Tangential Stress for Plane Strain (R = Ro, Ro = 3, B = 3).............................. 14 Effect of Tube Geometry on Axial and Tangential Stress for Plane Strain (R = Ro,, B = 3, N = 20)....... 15 Effect of Modified Biot Number on Axial and Tangential Stress for Plane Strain (R = Ro, Ro = 3, N = 20)......................... Page 5 25 27 28 29 30 31 33 34 35 36 37 38 39 iv

LIST OF FIGURES (Continued) Figure Page 16 Schematic Assembly Drawing of Test Apparatus.......... 42 17 Photograph of Test Apparatus.......................... 43 18 Experimental Temperature Data (z = 8, R = Ro, Ro = 3, B = 4.8, N = 18.9)............ 46 19 Experimental Axial and Tangential Stress Data (z = 8, R = R, Ro = 3, B = 4.8, N = 18.9)........... 47 20 Schematic of Modified Graetz Model................... 50 21 Schematic of Modified Leveque Model................... 54 22 Schematic of Physical Models Used in Determining Effect of Axial Conduction............................ 57 v

NOMENCLATURE AM coefficients for modified Graetz solution CL Helmholtz function B modified Biot number ~Bm coefficients for modified Graetz solution (5 3parameter including modified Biot number C constant deformation specific heat C constant in modified Leveque solution C-~ constant pressure specific heat of fluid ~E ~ modulus of elasticity e cubical dilitation G shear modulus ~h ~ heat transfer coefficient h overall heat transfer coefficient Ir ^-modified Bessel function of first kind of order v 3y Bessel function of first kind of order v Kr modified Bessel function of second kind of order /y kk+ thermal conductivity of solid and fluid, respectively M[, coefficients in particular stress solution N modified Nusselt modulus /nY integer Pe Peclet number Yr 7a/ radial and axial heat flux, respectively R dimensionless radial coordinate PRo dimensionless outside radius vi

R9 radial body force 6<k eigenfunction r radial coordinate < [inside radius outside radius S Laplace transform parameter AL entropy r tube-wall temperature To fluid temperature To inlet temperature Tc ambient temperature t time U average velocity LA radial displacement Uc. Up complimentary and particular solution for &., respectively Wt internal energy Atc local fluid velocity V displacement vector U-r axial displacement uc )u p complimentary and particular solution for u-, respectively Yr Bessel function of second kind of order /transverse coordinate eZ dimensionless axial coordinate "Z~ dimensionless axial coordinate axial body force ~t yaxial coordinate vii

r d sr, CI ) E~ 91 ) EL Ci, 8 9-r, SE, e9 -X4\ r 4.?r 4, Tp'F thermal coefficient of expansion Gamma function percentage deviation in temperature shear strain arbitrary end parameter radial, axial, and tangential strain components axial coordinate transverse coordinate dimensionless fluid temperature dimensionless temperature dimensionless temperature dimensionless fluid temperature transformed dimensionless fluid temperature dimensionless fluid temperature Boussinesq-Papkovich displacement potential separation parameter eigenvalue eigenvalue parameter Poisson's ratio transverse coordinate density of solid and fluid, respectively radial, axial, and tangential stress components, respectively dimensionless radial, axial, and tangential stress components, respectively shear stress dimensionless shear stress viii

i) dimensionless tube-wall temperature WX yLove-Galerkin displacement potential IP Goodier displacement potential ix

CHAPTER I INTRODUCTION The basic equations of heat conduction and thermoelasticity are well known and have been in existence for a long timeo However, the number of existing three-dimensional solutions appearing in the literature is very small and most of these have been published recently. Comprehensive bibliographies have appeared recently in the texts by Nowacki,(11) Parkus, 3) and Boley and Weiner.(3) Many of the three-dimensional solutions deal with the axisymmetric distribution of stress in cylinders. Two of major interest are mentioned here. Youngdahl and Sternberg(l8) obtained an exact quasistatic solution for the thermal stresses which arise in an infinitely long elastic circular shaft when its surface temperature undergoes a step-change over a finite band. The surface temperature over the remaining portion of the cylinder was held constant. The thermal stresses (14) in pipes were investigated by Parkus.(4) The case of a hot liquid in steady flow through a long tube transferring heat to the surroundings was considered. The inside and outside heat transfer coefficient was assumed to be large and slug flow was assumed to exist in the pipe. It was pointed out in these investigations that the assumption of infinite heat transfer coefficient is unrealistic but will lead to a conservative estimate of the stresses in an actual situation. However, it is often important that more realistic values be obtained since in many practical cases the conservative result greatly overestimates the eactual stresses It is therefore desirable that more realistic physics be included in the formulation of the problem. -1 -

-2 - It is the purpose of the present investigation to consider a more physically meaningful model to Parkus' problem. The thermal part of the study is based on a modification of the classical Graetz problem in convection heat transfer by including the heat capacity of the tube wall. (For a review of earlier work on the Graetz problem see for instance, Knudsen and Katz(8) or Jakob(7)). It is assumed that heat is transferred to the ambient through an overall heat transfer coefficient which includes both the thermal resistance of the wall and the convective film coefficient. The modified Graetz problem described above was studied by Schenk and Dumore. Use of the separation of variables led to an eigenvalue problem of the Sturm-Liouville type. Numerical results were given for three values of the finite overall heat transfer coefficient and the first three terms of the series solution were given. The results of Schenk and Dumore are extended to include six values of the overall heat transfer coefficient. In addition the series solution is extended to practical limits using numerical computation procedures. Since the series solution near the entrance is slowly convergent an approximation based, on a modification of the classical Levenque problem (see for instance, Knudsen and Katz()) and valid in a small region near the entrance is obtained. For the thermal stresses associated with the problem it is common practice to estimate these stresses by the plane strain approximation. In this study the exact axisymmetric stress state is determined and the validity of using the plane strain approximation is investigatedo

-3 - The general thermoelastic formulation for the axisymmetric distribution of stress in cylindrical coordinates is derived from basic principles. The approach used gives physical significance to the various terms in the formulation. For steady problems the formulation reduces to the uncoupled quasistatic theory. The thermal stresses are found in terms of Goodier and. LoveGalerkin displacement potentials. On the cylindrical surfaces the boundary conditions are satisfied exactly while at the entrance the stresses are self-equilibriating and Saint Venant's principle applieso Representative numerical results are given for surface temperature and stress. Experimental verification of the analytical results was obtained and a limited amount of data is presentedo

CHAPTER II THEORETICAL ANALYSIS A. Formulation 1. Thermoelastic Formulation of Axisymmetric Problem The general axisymmetric problem is consideredo The equations of motion and thermodynamics are applied to a system. Then the compatibility and constitutive relations are introduced. Finally the steady thermoelastic formulation is obtained. Application of Newton's second law of motion to the system shown in Figure 1 yields r r y - - R =f r^ (1) a +o, ) + z A t (2) r r4J where Gr denotes the radial stress, A- the axial stress, 0~ the tangential stress, ' the r-y shear stress, r the radial coordinate, e the axial coordinate, t the time, p and 2 the radial and axial body forces per unit volume, respectively,, the mass density, l. the radial displacement, and wr the axial displacement It is noted that since no circumferential distortion occurs the r-ed and 0-k shear stresses are zero. Furthermore, moment of momentum requires that -ru tr - T Applying the first law of thermodynamics to the system yields P tZ l 1 l 2 - J t (r!~ - {- a (G r} + L I (rr ^/)+ l rr ) + 2 - + 7r (3), -t+ 31 at / ^r odr t r Dr d Ct t (

z (oC + az dz) rd(dr z az T + a1 dz) rd dr ar rd(dz (rT + a- (rT)dr) d(dz ar d ra + a Crcy) dr) dodz r r a I T rdddr a rd dr z Figure 1. Schematic of System.

-6 - where A denotes internal energy per unit mass, Qr the flux, and?y the axial heat fluxo Multiplication of Equations (1) and (2) by t respectively, yields relations expressing conservation of energy. Subtraction of these from Equation (3) gives the expression for the conservation of thermal energy: radial heat and At ) mechanical following f Lk - -1 2- - i o-r rat dr a~t r _t r at c) 'LA-:za D +;;aat 9L LA+ -L L (4) Here the internal energy may be expressed as a function of temperature by considering Lk - aC (, 7 E, eV, ) where T denotes temperatures, r the radial strain, p the circumferential strain, E the axial strain, and Y the r- shear straino The total differential of k is;a~i dl- i;Ze d 6 Cl D~~~ C') y o, & - 4- C- 0 ~ E ) y - - _~ dE q + F j6-i +/\ d t~ a i -4 -r 6 ", 6 -Z Y 36 - -rl I- & 9 ) T rV 64)6-.I (5) In terms of the thermodynamic property relationship (which is valid both for reversible and irreversible processes) da = Td h + 8 ( -- TcK + i-( 8 + d e a c td) (6) where 4 denotes the entropy per unit masso Equation (5) may be rearranged to give

-7 - -d -- c- r,, d~T rlT~~~ Tl3 t 6 E 6 - I i - + ) ' + LF - t T9 dG-t- + TT(,s1r14,, 6 i e] de. 3' )T e v - C-I Z. -L t 8 (7) where C 1^, P e:, is the specific heat at constant deformation~ To eliminate the entropy from Equation (7) the definition of the Helmholtz function, o-, is required and its differential is given by do- -= +( fr a ^ + dt +- 3d.fe tdK') - r. (8) Noting that the differentials appearing in Equation following Maxwell relations suitable for solids are (8) are exact the obtained: 1 \ lttlT,eyr,&, - )6 T, \ i- 6It - -) % E k3^ J Ti 6ri Eq I 6.* f T sr ) 6,, 6t, y( -7 r JlT), 6(p e,, ' \;T = erip,& 6. (9) Finally substitution of Equation (9)into Equation (7) yields dl. =CdT I -T(- t + -j d6r + [-T(T4q) + O 7V d + 7[-T(tff) Z]d 8 dEt + i [T-rT )+ I d (10) where subscripts have been omitted but evaluation of partial derivatives at constant deformation is impliedo

-8 - The strains are related to the displacements in the same manner as in isothermal elasticity since purely geometrical considerations are involved; the pertinent equations are as follows: Usu --- - r T = d==+ ar (iiy Substitution of Equations (10) and (11) into Equation (4) yields the following form of the energy equation: tfC - T T w - X at + ar at $+f arit ] = — r- r (12) To complete the formulation constitutive relations are necessary to furnish a sufficient number of equations among the dependent variableso The relationships given by Fourier's conduction law and Hook's generalized stress-strain law are?r = - = ar ^ =- - D K T(13) and e = [(r -k-( +^;^ + AT emp = zIlQ - ^(1 -Ktr% + PT En =- r - v( ^ go) 3 + ^'r & I e + (14) I~

or solving for stress -~' 2G ( & + - e _ ' ' i1T ') 4 = i( ( + z e -, PrT ) e - z7, (" l + V /+Vr' r =- c ~ (15) where k denotes the thermal conductivity, E the modulus of elasticity, 2r Poissons ratio, 5 the thermal coefficient of expansion, and C the cubical dilatation ( e - Cr-+-p +&- ) - Making use of the constitutive equations, the coupled governing equations are written in terms of temperature and displacement as follows: V'u _ F t r - a t (16) Vr aG I- ( A) ) (17) Er) T ae r Tr (r ar \ (k (18) where e-=V and V denotes the Laplacian in cylindrical coordinateso Various simplifications of the coupled theory are discussed by numerous (3) authors, e~go Boley and Weiner(3) In most commonly encountered problems the effect of mechanical coupling and inertia is negligible, Neglecting these effects Equations (16), (17) and (18) become TV> - d t Tr r - 2 ( o),r t ^ = d (19) t- hV - + ay -,- t (20) CT = r a r) (21) Cc_ )t r ar Dr/ t 1 ~(l

-10 - respectively, which constitute the uncoupled quasi-static theoryo Most problems are solved using this theory. For unsteady problems the time variable then appears only as a parameter in Equations (19) and (20) and therefore simplifies the problem considerably~ For steady problems time disappears altogether and the coupled formulation reduces to the quasi-static formulationo The following analysis applies to the quasi-static theory and non-thermal body forces are omittedo By virtue of the linearity of the problem the solution of Equations (19) and (20) can be written as the sum of two solutions UL - kp + LtAc krj- ur + ir (22) where the p subscript denotes a particular solution and c denotes the complimentary solution of the homogeneous equationso A particular solution is given by Goodier ) in terms of a scalar displacement potential which is defined by HiJ^~~ _-~~ it^~ ~(23) The Goodier potential then satisfies the relation oY I / ( v ) \ RT (24) which is obtained by substitution of Equation (23) into Equations (19) and (20). The complimentary solution is given by use of the third (10) component of either the vector displacement potential of Love Galerkin(5

-11 - 7 17 x J; =L Lc-wy v17' - a (25) or that of Boussinesq ( —Papkovich(2) ucc W- L C->1 ) V\A_. - ) (26) These potentials satisfy the biharmonic and harmonic equations V 7V =7 ~ (27) V_^. = 0 (28) respectivelyo Finally the stresses may be written in terms of potentials by combining the particular and complimentary solutionso In terms of Goodier and Love —Galerkin potentials the stresses are -r S ZL (-1U -v4W) + a(Io7 -- ) - = zL (tar -v') ~+ -^ r )] T ( 2z -r) + r(u-vJv [ —r) (29) dra~~~d aal/ J.(eY Vzs), and in terms of Goodier and Boussinesq —Papkovich potentials they are

-12 - a(+ —~ 2~1D +~ -- _ X, - a[ (1; - VL*) + zV Af - 3 ar L L(e - V7"l#) + (.-^ -3 _] 24 = A + (I-L/) - - L?, (30) 2. The Temperature Problem The following assumptions are made: 1) Axial conduction in the tube wall and fluid is neglected. The effect of axial conduction in the wall is discussed in Appendix C. Schneider(16) investigated the effect in the fluid and showed that the effect is negligible when the Peclet number (Reynolds number times Prandtl number) is greater than 100. Because of the low Prandtl numbers of liquid metals (.003-.03) axial conduction may become important for this class of fluids. 2) The physical properties are constant. 3) The fluid has a fully developed laminar velocity profile. 4) The external convection heat transfer coefficient is constant. 5) The fluid temperature is uniform at the entrance. 6) The ambient is isothermal. 7) The fluid is incompressible. A schematic of the physical model is shown in Figure 20. Under the assumptions given above the governing differential equations are derived in Appendix Ao The resulting dimensionless formulation is given by

a ( R R ) - (31 Pet(-Rv ) - I A (R 2 ) (32) subject to the boundary conditions I ), ) = ) ( ) (33a) BROi) _ _ — 1 (R ) (33b) R oo ) - I (33c) _ /, R ) = (33d) ajt/1,) = _A ^/l) J(33e) where R and. denote the dimensionless radial and axial coordinate9 respectively, ( and 6 the dimensionless temperature of the wall and fluid, respectively, Pe the Peclet number, B modified Biot number, and N modified Nusselt numbero Bo Solution o Temperature Solution The temperature distribution in the tube is determined from Equation (31) subject to the boundary conditions Equations (33a) and (33b)o The simple solution is readily obtained as -pR ) = L I - JR (Cl, ) (34) where 13 - + a R o

-14 - Determination of the inside surface temperature requires solution of the convection problem. The solution of the convection problem is found from Equation (32) subject to Equations (33c), (33d), and (33e). Applying the separation of variables a product solution is assumed in the form G - ^cW i(po (35) and substituted into Equation (32). This yields the two ordinary differential equations t Pe. = D (36) ( R 8R) t >-R(R - Ri) 0 (37) where \ is an arbitrary real constant of separation. Substitution of Equation (35) into the boundary conditions, Equations (33c), (33d), and (33e),yields tCeo O(R) "-1 (38a) dS^ ~ A(~)~~~'~ - 0 ~(38b) RC) = - N (o) (38c) The solution of Equation (36) is well known and can be written immediately as - t < i) = — C, e P' (39) where C, is an arbitrary constant. The solution of Equation (37), however, cannot be expressed in terms of previously tabulated functionso

-15 - Therefore solution requires computation of the eigenfunctions and eigenvalues of the system which is of the Sturm-Liouville typeo A power series solution of Equation (37) can be written in the form ( = 2 c_ R (40) v=0O where C.nd-l = M0 and Q2, - - (2- ) ) The characteristic values, y, are then the roots of the polynomial obtained by substitution of Equation (40) into Equation (38c)o However, in practice, only the first few roots of this polynomial can conveniently be determined. Instead of using the power series method of solution a more direct numerical procedure was used in the actual numerical computationso The differential equation, Equation (37), was integrated numerically using the Runga-Kutta methodo Eigenvalues were determined with a successive approximation procedureo After integration over the domain for a trial value of >A appropriate values of (CO) and R were substituted into the boundary condition, Equation (38c)o For the correct value of At Equation (38c) becomes an identity~ The eigenfunction was determined by a final Runga-Kutta integration using the appropriate eigenvalue Having determined the eigenfunctions and eigenvalues the general solution of the convection problem may be written as,4-t =_ X=0 2 -QCRt ) r fi, C(f ) P (41) ^\-SO The ~ are constants to be determined in such a way that the remaining

-16 - non-separable boundary condition is satisfiedo Substituting Equation (41) into Equation (38a), multiplying through by R (-R ),, and integrating the resulting expression from 0 to 1 yields ~ I (l- R ) (42) It is noted that the ^, are orthogonal with respect to the weighting function R -R') on the interval 0 to 1 o Successive application of Simpson's rule is then used to evaluate numerically the integrals appearing in Equation (42). Finally the temperature of the fluid at the inside wall is given by &Bio ^ = 2 L1r30, E (43) where =, ( (I) and - Values of the eigenvalues, Am, eigenfunctions at the wall, (,,O), and coefficients, F, and an, are tabulated in Appendix D for N= 20, 17, 6, 2, 1, and 05. It was found feasible to carry the computations as far as the first 20 terms of the serieso All computations were performed on an IBM 7090 computero Comparison of the numerical results was made with those of (15) Schenk and Dumore for the first three terms of the series Good agreement was found for the eigenvalues and eigenfunctions while some of the coefficients showed minor discrepencieso It is noted that the nomenclature in their paper is related to the notation used here by e =z 2- and NL A N o

-17 - Solutions obtained by separation of variable s and Fourier series are "large time" solutions and thus slowly convergent for "small time"o Therefore an asymptotic solution valid at the entrance is obtainedo Based on the physical model shown in Figure 21 the governing differential equations are derived in Appendix Bo The resulting dimensionless formulation is given by Y = a _ (44) subject to (. 0,) D) - 60 (45a) _-(.m o, ) - 0 (45b) -C;) N /[ -(oV+o ) - (45c) where -- denotes the dimensionless fluid temperature, y and Z the dimensionless transverse and axial coordinates, respectively, and N a modified Nusselt number A solution may readily be obtained by the use of Laplace transforms. Taking the Laplace transform of Equations (44) and (45) with respect to the dimensionless axial coordinate A yields d + ( ya - Sy-(, ') -0 (46) subject to the boundary conditions 3 (co s ) 0-= o (47a)

-18 - d- ( 0,^ s) /_ (os) - S (47b) where S is the transform parameter and 8(:S) the transformed temperature The solution of Equation (46), noting that At;,, ) -, c9 may be written as — 5 ) = i y" k^(3 s" t2) (48) which satisfies Equation (47a). The functions Jr-.(X) and K,(x) are the modified Bessel functions of the first and second kind, respectively, of order ir and argument X o The constant A is found by substitution of Equation (48) into Equation (47b) and is given by Cf - S'/ l ( C S /3) 19)(49) where CI 3z C 3' l() - and P(X) denotes the Gamma function of argument X Having obtained a solution of the transformed formulation the transformed wall temperature is obtained by taking the limit of Equation (48) as - approaches zero The result is DsC ) --- s5,(,Cs-'0 (50) where _ N,l (') Since a solution which is rapidly convergent in the entrance region is desired, a "small. " solution is obtained by noting that large values of the transform parameter S correspond to small values of F o Consequently, the expansion of the transformed solution into

-19 - a series of ascending powers of s and subsequent term-by-term inversion gives a solution useful for small values of ~ (see for instance9 Arpaci(1))o In the above) P(s) is a function of S whose form is determined by the particular form of the solution under studyo Before proceeding recall that the sum of a geometric series is given by -- s) - I + fC(s) +- (s)-. C+ (51) Choosing -(s) = - S and substituting into Equation (51) yields +t+cs-s3) = - (C s-3 + Ccs-"3) -. (52) Substitution of Equation (52) into Equation (50) and rearrangement of terms then produces - os) c= ( s - C s- ' + - +c' ) (53) Referring to published tables of Laplace transform pairs the transformed solution can now be inverted term-by-termo The wall temperature is therefore given by -t3-o, C) = 2, (-1)-/( \ Referring the wall temperature to T, instead of To, Equation (54) may be rearranged to give W-f (o, -) - I + E. _ _(o r (55) were t T (+ vc) were -9T- T Ic T Ic0

-20 - 2. Stress Solution A particular solution is found using Goodiers thermoelastic displacement potential by examination of the particular form of the temperature expressiono Once the particular solution is found a Biharmonic Love-Galerkin function is obtainedo For convenience the exact temperature distribution in the wall is written here as pCR^t) = ( I-B R L 8MA Re t (56) By inspection of Equation (56) one can construct a particular solution to Equation (24) It can thus be verified by direct substitution that (s = 1 C( I - J 4 R ) c (57) is a particular solution of Equation (24), where /I is the Goodier potential corresponding to the n-th term of Equation (56) and = =- (JV ^) ^ 0^ o By adding all of the particular solutions of the form Equation (57) one obtains tv = U(- B^CAlR) ^JL Me e~^lt (58) One can obtain a solution of the biharmonic equation by writing Equation (27) as a system of two second order partial differential equations given by 7'A =7 (59a) V' = o (59b) where 4 is an intermediate functiono (see for instance, Arpaci (2))

-21 - Using separation of variables and noting that L" Q-t P- the solution of Equation (59b) being well known is written immediately as K (R ) = ( t Ry) e- (60) where 3, and Yo are the ordinary Bessel functions of the first and second kind, respectively, of zero order and,, is the separation parameter which was defined in the temperature solutiono Substitution of Equation (60) into Equation (59a) yields IV X ( A,-i - (61) Since the right hand side of Equation (61) is composed of product terms a convenient form of the Love —Galerkin function can be written as Dy = (=- /JoC- (,OI ) R X, (MaRJ) e ant (62) (62) This relation was chosen because f (R,(4 ]R~) ese" are known to be \R ^. (vYR) M biharmonic functions Another approach to determining %C is to assume a separable solution of the form XC(R,) = FLR) e-' and substitute into Equation (61)o The complimentary solutions of this equation are Th o(AR) and Yo0([R) and a particular solution may be obtained by variation of parameters. However, it is expedient to return to Equation (2) and proceed from there A solution of the biharmonic equation in terms of four arbitrary constants cc,, b, c, and de is written in the form Xm -,- I1(.,,R) +- b, Yo ) -t RIC RJ,(AR) -t dm(yR )] e (63)

-22 - where the /r subscript refers to the, ~th term corresponding to the particular solution. Summing over all terms one obtains X = 2 [ (AR^,, + b(YP) +,RJ, (,R) PdRY, )1] A (64) The stresses are obtained by substitution of Equations (58) and (64) into Equation (29). After expanding and combining terms the stresses in dimensionless form are given by t7 = {((f0) f R] B -1)B - a, [co ) 4M 0 _-, 3, e) I - K [ Y, (,-R) A- R Y (.R )l + It O (t- t) D,(^ f) - MI R 3)(M1R)2 + d^7, C (-z) Yo - az - > CR) )} Cr)] F e (65a) Q~ 0- [i(, z*(R)I 3- by [ tR X (nR) ] cn { MC CUo l>,> 4 bCY/,o^^ l + /c, ^, R J (vF R) - L(z- Z) 3ogX^- ] t^ dC8 Ryl(, PR)- z(zv) y ] 4R) ] e (65 ) r = Zx t x I 1 B3, + Qe a U[(stR) ] $4 btY.(XXy, R) tcl ^ raR J;(oRB) + Z((-Y) J. l(cR )~ - d^, [ R Yo(*, R) + r( -O )/ Xr^) t} e (65d)

-23 - where the general dimensionless stress is related to the actual stress by ~ -- - being the actual stress. z2 CTo-C;? (., —c-), Since the surfaces are free of externally applied forces or restraints, the boundary conditions are Cr c t, - 0 'r (Ro,0)- 0 r (i,) -D r (RR ) == D (66) The stresses at i = 0 are self-equilibriating and Saint Venant's principle applies; for -ev all stresses vanisho Finally substitution of Equations (65a) and (65d) into Equation (66) yields the following four equations in the four unknown coefficients: L 3o (a% ~~ 7^ J1(/^Al I ^Q - C o(L -" JM Y. I^ J b( L (8-y ^^Jo ) (r - 3- (^< ] /c^ ~[(/-,,y, (, ) - -^,(^M]} = o -C - ]B (67a) -L loo^^4RO - *~^R<> tsiPo] CL ~ YI NRoC^) - ao)sRcl b, + 4 (I- tf) 3o (AMR) -, RV J R "( l>) XCt + [ ( - /^Z) Y (uf c) 0Ro%/i(^j ) ] d - { [ (Rt )+ R.1 B - II} 6K (67b) C Ji (M^) ' CL +- C Y. ](>r b^ -C ~Y, 3o c^^ a L(i- ) ji )^} ] /^^ -[A4~y.^,^ z-I-T) Yt ^ ] d - [ 6^ (67c)

-24 - [a, iuln 1] + t yK, Ro) b+ |[ >R.o Jso(a>R + t(l-() \ j(^R~) 1 ICO -[| Pao y R) ( q i l+ R= /ro (67d) The unknown coefficients may be determined by application of Cramers rule and the theory of determinantso However, standard matrix methods of evaluation were used in the actual numerical computationso C. Results Representative numerical results are given in the form of graphs and the effect of the various parameters is illustratedo The parameters appearing in the solution are Pe, the Peclet number (Reynolds number times Prandtl number), NJ, a modified Nusselt number, &, a modified Biot number, and Ro, the ratio of outside to inside tube radiuso The natural parameter in the temperature solution is the dimensionless distance - which is the inverse of the Graetz numbero The Peclet number appearing in this parameter may vary from a high of approximately 107 for viscous oils to a low of zero. However, because of the assumptions used in the formulation the solution is useful for Peclet numbers down to approximately 10o The temperature distribution at the inside wall of the tube is shown in Figure 2 and the effect of modified Nusselt number is illustrated. Values of N may vary from 0 to x o The limiting case of N = 0 corresponds to an insulated wall and N = o corresponds to a constant wall temperatureo In this figure as in succeeding ones the

1.0 4 I I R =i w w -e3: LU w -J c-J 0 U) z w 0. 0. 0. 0. 80 610110 1 I |] 0 ~I I |11H111 1 1 [ c,4[I 111111 SDII '\l~~l I IllLo11111 R=1~~ 6-W11111E X Itt1:11 E; hi [ PT~~fi0 -:!; b~~f11110 X~S2 I PQ Y1 10-6 10-5 10 10-4 10 103 10-2 10-1 10 100 11 DIMENSIONLESS DISTANCE, Z/Pe Figure 2. Effect of Modified Nusselt Number on Wall Temperature I. (R = 1)

set of curves for small h was obtained from the approximate solution and the set for large p — was obtained from the exact solution. In Figure 3 the inside wall temperature is plotted as a function of the dimensionless axial distance, Z, for the same values of N The Biot number is a measure of the relative importance of the internal thermal resistance of the wall and the external film resistance. As the Biot number increases the relative importance of the thermal resistance of the wall also increaseso Values of the Biot number may range from 0 to cO. Figure 4 illustrates the effect of B on outside wall temperature. Finally the effect of the wall thickness ratio on outside wall temperature is shown in Figure 5. The temperature drop across the wall increases as Ro increases. The maximum curvature of the axial temperature profile occurs near the entrance of the tube. Thus the main effect of the axial curvature on the stresses is also confined to this region. In the stress results which follow the plane strain solution is plotted along with the exact solution for reasons of comparison. In Figures 6' and 7 the axial and tangential stresses at the surface are compared. The stresses behave in a similar manner and the axial stress at the outside surface deviates most from the plain strain case. Therefore, only the effect of the various parameters on the axial stress at R = Ro is shown in the following resultso The radial and shear stresses along the coordinate directions are generally much smaller than, the maximum axial and tangential stresses and are not giveno

e = I -P - = ~ t.~ K \ |L6 ^____e_______ — u ~ 0- 0ia 620 I 6 5 4, I, t. N "$ DISTANC rexpertre II^ ------— ~of lified lu sselt Itbe '..Effect = lo0) Fi.gure 3' ( l, Pe

1 C 0 - & LJ I0: -J 3: C/) LLU -j z 0 z LU 2: — 4 0. R = R II -— r 1 N =- 20 8, ___ ___E__ 1 —_X__=_:_=l 0.3 0. N 1. 0 " Nb I I I ~ ~ ~ ~ b ~ b % I N N bft I 0.: 2 3.0 IQ Fin- 10 — a em I 10-6 1-1 10 10~ DIMENSIONLESS DISTANCE, Z/Pe Figure 4. Effect of Modified Biot Number on Wall Temperature. (R = R, Ro = 3, N = 20)

1.0 I I I I I I I I.......!I!I I I II I I 1 [ i IiI I I I Elf R=R R = RO,______30 B = 3 1~~~~~~~~~~~~~~......., 8 N = 20 0. -eLLI aL,) Q. 21 CL) -J 0 Z: z w LU s4 5z I L R = 1.0 _ _~~~~~.II I N 0.6 N - - 0..1 1^2 ' 4 ____ ____,________ I - I _ ~,,, 1.,,, ___I,__,,_1_,___-_r_, 2 --- ^ ^ -— 3.U ~ ^ ^ ^ ^ --- —--------------- I 0. -4 10 1 - 10~ DIMENSIONLESS DISTANCE, Z/Pe Figure 5. Effect of Tube Geometry on Wall Temperature. (R = Ro, B = 3, N = 20)

-30 - -.24 *K -K ti o -.20 0, UL CO iw cn en -.16 cu> LU _J 0 1-4 W I'm 12 -.08 -.04 0 0 1 2 3 4 5 DIMENSIONLESS DISTANCE, Z Figure 6. Comparison of Tangential and Axial Stress I. (R = 1, Ro = 3, B = 3, Pe = 10, N = 2).

-31-.10 ---- EXACT.09.08.07 -e0 -I N. 06 (n LU) rw LJ Z.04 -.03 CO Uo.03.02 -.01 0 0 PLANE STRAIN R = R R =3 O B= 3 Pe = 10 N= 2 z* = f * az*a Z 1 2 3 4 5 6 7 DIMENSIONLESS DISTANCE, Z Figure 7. Comparison of Tangential and Axial Stress II. (R = Ro, Ro = 3, B = 3, Pe = 10, N = 2)

-32 - The effect of tube geometry is shown in Figures 8 and 9. As the ratio of outside to inside radius increases the axial effect increases at constant dimensionless distance. Figure 10 shows the stress behavior for various values of the modified Nusselt numbero In the limiting cases of N = 0 and N = t the thermal stresses are zero (neglecting end effects) since the tube wall temperature is uniform. Therefore maximum stresses are obtained for intermediate values of No Figure 11 shows the effect of Biot number on the stresses. For B = 0 the radial gradient is zero and the stresses are the result of the axial gradient. The axial effect decreases as B increases. Finally the influence of the Peclet number is illustrated in Figure 12. In the plane strain solution the axial variation is carried as a parameter. Thus the plane strain stresses vary as the temperature varies. Figures 13, 14, and 15 illustrate the behavior of the plane strain solution and show the effect of modified Nusselt number, geometry, and modified Biot number, respectively. The plane strain solution is given in Appendix Eo In the numerical computations the number of terms required for convergence of the series solutions variedo Computation was terminated when the last term of the series added was less than.05 percent of the sum. For the exact solution the number of terms required increased as decreased and all 20 terms available were used for ~ of the order of 5000. The exact number of terms used varied with N, the larger values of N requiring more terms for a given -e o For the approximate solution more terms were required as increased and up to 50 terms of the series were used.

-33-.10 -EXACT.- -- PLANE STRAIN ic N O U) uJ On U) U) U) I CO LJ z 0 U) z LU Q.09.08.07.06.05.04.03.02 R= R B = 3 Pe = 10 N = 2 \R = 1.6 \\ 1.2.01 0 0 1 2 3 4 5 DIMENSIONLESS DISTANCE, Z Effect of Tube Geometry on Axial Stress I. (R = Ro, B = 3, Pe = 10, N = 2) 6 7 Figure 8.

-34-.10.09... — EXACT -- - PLANE STRAIN.08.07 R B Pe N = R = 3 = 10 = 2 N 0 () Cl) LIJ U) U) U) Ll -1 z vo 0 U) LU 5-4.06.05.04 = 3,1.6.03.0.0 0 1 2 3 4 5 6 7 DIMENSIONLESS DISTANCE, Z Figure 9. Effect of Tube Geometry on Aixal Stress II. (R = Ro, B = 3, Pe = 10, N = 2)

-35-.20.18 - EXACT _ — _ PLANE STRAIN.16.14 RR = B = Pe = R 1. 1.6 3 10 4C N 'O U) LLI CC (I) l) (I) Lli -j 0 z w N = 0.5.12.10.08.06 1.04.02 0 1 2 3 4 5 6 7 0 DIMENSIONLESS DISTANCEI Z Figure 10. Effect of Modified Nusselt Number on Axial Stress. (R = Ro, Ro = 1.6, B = 3, Pe = 10)

-36-.14.12.10.08 ic N O C( U) IU) LL LLJ. q.06.04.02 0 -.02 -.04 0 1 2 3 4 5 6 7 DIMENSIONLESS DISTANCE, Z Figure 11. Effect of Modified Biot Number on Axial Stress. (R = Ro, R = 1.6, Pe = 10, N = 1)

(9 = m 'E = a '9'T = o~ o'I = a) 'ssasBS TITxV uo jaqumnK aeaVog Jo o@aSjSJ - 73T Gam2TJ z '33NViSI SS31NOISN3WI S c Z L 9 T 0 0 -01:0 ' 0s O' m 0 z rI SO' uC 0o C,;, s Q N LO' 80 -60' OOZ = ad 9 = N ~ = g 0 9o = H = H NIVi.S 3NVld 13VX3 OT

0.25 * -- 0.20 b 1C N U) CO nw U) U) w LLJ lCO 0 U) _4 z Q.. 0.15 0.10 N=.. 5 PLAIN STRAIN R=RR B=3 20 X ~ \ I I \ Io-I ____I____ \.1________ 1 ____\ \ I_____ I === =| |I-= I 0o 0.05 0 -6 10 10-2 10-1 100 101 DIMENSIONLESS DISTANCE, Z/Pe Figure 13. Effect of Modified Nusselt Number on Axial and Tangential Stress for Plane Strain. (R = Ro, Ro = 3, B = 3)

0.25 * -0.20 0.N on o 0.15 LLI crFc0 0n -J 0.10 Z: 0 C/) w 2: m 0.05 ______PLAIN STRAI _ R=R S 1 R = 3,0 |~ B=30 --— ' -------— N-70 - 1.1...._1 ________I _ "~^ I \^... _ — _.. I ^ — ^ \____________I______________I__,___,____1__ I \O I 0 10-6 10-2 -1 10 DIMENSIONLESS DISTANCE, Z/Pe Figure 14. Effect of Tube Geometry on Axial and Tangential Stress for Plane Strain. (R = Ro, B = 3, N = 20)

0. A 41 — 1- - -I — -1 1- - - I - - --- - I-. - - - - z~~~~ i, ' i i lr [ ~ I... -W I I I ~ I I i I I I I I i I I I I *:t * O. cn 0 N C) LLJ IJJ L) 0. C/) C,, L_ Z 0 C/) z 0. I.,. PLANE STRAI 3 R = R:" i^ N = 20 2 0. n. ------ ~3~^ ^ o I -6 10 10-1!0Q 0 10 DIMENSIONLESS DISTANCE, Z/Pe Figure 15. Effect of Modified Biot Number on Axial and Tangential Stress for Plane Strain. (R = Ro, Ro = 3, N = 20)

CHAPTER III EXPERIMENTAL INVESTIGATION A parallel flow tube and shell heat exchanger was constructed and experimental values of temperature and strain on the outside surface of the tube were obtainedo Hot and cold tap water was supplied to the tube and shell side of the heat exchanger, respectively. For the lower flow rates the hot water was supplied to the test section from a head tanko A type 321 seamless stainless steel tube with.750 inch OoDo and.250 inch wall thickness was used as the test pipeo Two and one half feet of the tube extended upstream of the heat exchanger and served as the velocity development sectiono A schematic assembly drawing of the heat exchanger is shown in Figure 16. Figure 17 is a photograph of the test apparatuso At the upstream end of the heat exchanger the header chamber was silver soldered to the pipe. On the downstream side a sliding o-ring seal allowed free axial motion of the tube relative to the shell. Copper tubing with one inch IoDo and beginning 3/16 inches from the plastic insulator was used for the shell, giving a clearance of 1/8 inches in the annuluso Temperature sensors and strain gages were mounted on the outside surface of the tube with high temperature cement and were installed at the axial locations indicated in Figure16o The temperature sensors were of the nickel foil type (STG-50, Micro-Measurements, Inco) and the strain gages were of the self temperature compensating foil type (MA-09-062AA-120, Micro-Measurements, Inc.)o Lead wires of 30 gauge -41 -

COLD TAP WATER tI LASTIC I NSULAT m _ _ TO DRA I N STRAIN GAGE AND TEMPERATURE SENSOR INSTALLATION I r) I HOT WATER —X - - + I %-A -&"I IL 1~1~11111111lllll~fjlll 1 TO WEIGH TANK 1" Ne 47" 3" e-l --- J3 Figure 16. Schematic Assembly Drawing of Test Apparatus.

Figure 1'i iL. ehtogrelph of I tIest Appara'ttus.

stranded copper, vinyl insulated cable were soldered to tabs cemented to the tube. Jumper wires formed from one strand of the lead wires were then soldered to the sensor and gage tabso The lead wires were brought outside of the heat exchanger through two 16-wire Conax fittings mounted in the header chambero Because the installation was to be immersed in water the temperature sensors and strain gages had to be water proofed. Unfortunately application of a water-proofing compound insulates the surface to heat transfer and thus severe restrictions were placed on the choice of available materials. It was found that a coating of red Glyptal enamel over a coating of clear Glyptal varnish baked at the recommended temperature was satisfactory. In addition a thin coating of a rubber sealant was appliedto the solder connectionso The overall thickness of the water-proof coating and gage was less than e005 incheso The temperature sensors were connected to an LST resistance network (LST-100-120, Vishay Instruments, Inco). When used with an LST network the sensor-network circuit is equivalent to a half-bridge circuit with 120 ohm active and dummy arms. The output was measured on a model BAM-1 (Ellis Assoc.) bridge amplifier and meter. The strain gages were wired into half-bridge circuits consisting of an active and a dummy gage. Dummy gages of the self temperature compensating foil type were mounted on a stainless steel plate and imbedded in vermiculite insulation to eliminate thermal effects due to convective currents in the airo Using the 3 wire method of wiring to eliminate thermal effects due to temperature variations in the lead wires the half-bridges were connected to a multi-channel switching and

balancing unit (Baldwin-Lima-Hamilton Corp.). The output was measured on a model BAM-1C (Ellis Assoc.) bridge amplifier and meter with which a sensitivity of 1 microinch per inch per division was obtainableo At the beginning of the test series both the temperature sensors and strain gages were calibrated. The inlet fluid temperatures were measured with 30 gauge copper-constantan thermocouples and a portable precision potentiometer (Model 8662, Leeds and Northrup Coo) was used to measure the EoMoFo The mass flow rate was determined from measurements of mass and time obtained by use of a weigh tank and stopwatch. Data were obtained for Reynolds numbers ranging from approximately 400 to 50,000 with points in the laminar, transition, and turbulent flow regimes. The Prandtl number was approximately equal to 3 for all runs. Figure 18 shows the variation of outside surface temperature with Peclet number at - = 8 o It was found that the experimentally obtained temperatures were higher than predicted. The observed shift is attributed to the insulating effect of the sensor and water-proof coating. The experimental values of axial and tangential stress at E - 8 are shown in Figure 19. Natural convection effects apparently have influenced the results for the low laminar flow rateso The uncertainty in the temperature measurement was less than ~ 0.5~F and in the strain measurement less than + 2 microin/in for (9) the highest sensitivity used. Empirical correlations given by Kreith were used to compute the heat transfer coefficient to the ambiento

.25 w.20 cr L1 cL 0 -w.15 u1 _J.J 3~ I*.10 _J cn co 05 5.05 I I I I - THEORY * EXPERIMENT z=8 R = R RO=3 B = 5 N =18.9 BEGINNING OF TRANSITION I mono*, I., I c\ n 102 10 106 PECLET NUMBER, Pe Figure 18. Experimental Temperature Data. (Z = 8, R = Ro, Ro = 3, B = 4.8, N = 18.9)

* JU I U THEORY, a* = w w EXPRIMI * EXPERIMI NT, a NT, o 0 U -et) U) t- ~n U, -J z U). LIJ Z..,, in Ah Z= 8 B= 4.8 15. N= 1.9 __________ m ^10 _____ BEGINNING OF TRANSITION Ac- j _ II I I.U3 I n I 102 106 PECLET NUMBER, Pe Figure 19. Experimental Axial and Tangential Stress Data. B = 4.8, N = 18.9) (Z = 8, R = Ro, Ro = 3,

CHAPTER IV DISCUSSION OF RESULTS Examination of the stress results shows that the behavior of the axisymmetric and plane strain stresses is qualitatively similaro The effect of the axial temperature gradient is confined to a region near the end of the tube where the gradient is largest. At axial distances greater than three outside tube radii, approximately, the plane strain solution may be used to obtain the stresses since the difference between the exact and plane strain result is smallo Closer to the entrance the influence of the gradient may be important, depending on the values of the parameterso It is noted that the axial temperature gradient decreases as the Peclet number increases, holding the other parameters constanto Since the asymptotic temperature solution is a large Peclet number solution for dimensionless distance of order one, the resulting stresses deviate little from the plane strain solutiono -48 -

APPENDIX A FORMULATION OF TEMPERATURE PROBLEM A schematic of the physical model is given in Figure 20 The governing differential equations are obtained by applying the first law of thermodynamics to a system for the tube wall and to a control volume for the fluido Under the assumptions made in Chapter II the governing equations are ari (r - =- (A-l) fC UAi) - C(r J -r) (A-2) subject to the following boundary conditions T-(r ) = T (r, ) A-3a) -k T(r, h r(rO T (A-3b) T, (ro} - To (A-3c) f-_ =(- 0 (A-3d) -~f ~(rt ) ^- h T ri,r) - R e (A-3e) where r and ~ denote the radial and axial coordinate, respectively, rT the inside radius, r, the outside radius, r the wall temperature, T- the fluid temperature, T. the ambient temperature, To the inlet temperature, f the fluid mass density, C; the fluid specific heat, uL. r) the fluid velocity, t and K the thermal conductivity of the wall and fluid, respectively, ~ the heat transfer coefficient to -49 -

r I h,T0 INSULATION FLUID u*(r)! 0 z ' 0 Figure 20. Schematic of Modified Graetz Model.

-51 -ambient, and h I=- I[i no] the overall heat transfer coefficiento The fully developed velocity distribution in laminar flow is given by uc (n = wzUL i- ]r (A-4) where U denotes the average velocity. Substitution of Equation (A-4) into Equation (A-2) yields r cFCF u, -(L ^ P T - k r (r?r2) (A-5) _ r r Finally, rearrangement of the formulation into dimensionless form gives R ( R a =- (A-6) P. ( I- R) -a R dR y R /R (A-7) subject to w( I, t ) -= v(/, ) (A-8a);$. (RosiF) ^ = _- B (SP(RO^ (A-8b) (CR, D) 8"- I (A-8c) Si(0 i) =- (A-8d); 9 ( 1, ) ( - (A-8e) where the dimensionless variables and parameters are defined by

-52 -r r =- k _ - = r. 7f fi- _ Th - T 2 rL U z = )~ _- T -Tb Q = lk e- h ro H/ L } ) The parameter Pe is the well known Peclet number (Reynolds number times Prandtl number) based on diameter, 3 is a modified Biot modulus, and N is a modified Nusselt number.

APPENDIX B APPROXIMATE TEMPERATURE FORMULATION FOR THE ENTRANCE REGION A schematic of the physical model is shown in Figure 210 The assumptions applying to the formulation are: 1) The physical properties are constant. 2) The velocity gradient in the fluid is constant and equal to the velocity gradient at the wall of a circular tube with hydrodynamically developed laminar flow. 3) Axial conduction.is negligible. 4) The heat transfer coefficient to the ambient is constant. 5) The fluid is incompressible. The governing equation is found by applying the first law of thermodynamics to a fluid control volume and is given by suCL ) 2t - A (B-l) subject to l- (C A 0) = To (B-2a) Tf ( C, ^ ) = To (B-2b) ~ (0,) =_ = h [T(o.,) - 1 (B-2c) where ~ and 7 denote the transverse and longitudinal coordinate, respectively, T- the fluid temperature, To the undisturbed fluid temperature, T' the ambient temperature, h the overall heat transfer coefficient defined in Appendix A, Sf the mass density, C- the -53 -

Tf(Ez) FLUID I r 0 1 z h, Too Figure 21. Schematic of Modified Leveque Model.

-55 - specific heat, K. the thermal conductivity, and Ut (f) the local fluid velocity. The velocity distribution is 4 ( ) = 4r A (B-3) where U is the average velocity in the tube and r' the inside radiuso Substitution of Equation (B-3) into Equation (B-l) yields ___ _ 2fs G (B-4) Finally, rearrangement of the formulation into dimensionless form gives LfJt^ (B-5) subject to -9L,0o) =- 0 (B-6a) 89~C<~3 S) = 0 (B-6b) _ (_ 0) - 4 [-(oi) _ I 1 (B-6c) where T- -, r- -e( (B-7) p = ZrLL U ks _ _ I (_) _ _ r_ 3 and Pe and N are the Peclet number and modified Nusselt number defined previously in Appendix A.

APPENDIX C EFFECT OF AXIAL CONDUCTION IN THE TUBE WALL The effect is studied in the end section of a flat plate for (2) two cases as shown in Figure 22. In case 1, which was solved by Arpaci,(2) an exponential axial temperature distribution is maintained on one boundary while the other boundary is maintained at zero temperature. In case 2 both boundaries have the same exponentially decreasing temperature distribution. The axial effect at the end is shown as ( in Figure 22, Using the variational procedure the temperature profile in the wall for case 1 is assumed to be 8, = T S - 4 y O C -) XC4) (C-1) where 9B is the dimensionless wall temperature, /7 and ~ the dimensionless transverse and axial coordinate, respectively, a/ a constant, XZ() an unknown function of ~ to be determined, and ' is a parameter which satisfies Y. This restriction on Y is required in order that the approximate profile does not violate the physics and is determined by the end conditions, e.g. end fittings on pipe. The variational formulation is given by o'C?- t t+ D L0 d c d = /2 (C-2) Substitution of Equation (C-l) into Equation (C-2) and integration over the interval 0 - 1 of ~ yields lS /e - - X(43 )]SX J= (C-3) -56 -

uoT-olnpuoD TITXV Jo aoJJg UUTUTWm9I4aGC UT pasn slapo s To TSl1 d JO DOTuqS *- GaflGnSTJ T A 11-a 0 S 3SV3 0 0 T 3SVO

-58 - Upon equating the integrand to zero and rearranging one finds - /X e (c-4) where the boundary conditions are ZX () /j Z (Ib) = (C-5) The solution of Equation (C-4) being elementary is written immediately as X - 6 t- l) t t- Q' -_ 8 ^)De-i (C-6) Finally, substitution of Equation (C-6) into Equation (C-l) yields - B4 -e(,-z)[b e- t e ( )e-^-e (C-7) For case 2 the temperature profile is assumed to be h9= e4St - 4y~i-t) Z(0 (c-8) where the previous restriction on ~ is not required. Following the same procedure as for case 1 but omitting the intermediate details one obtains ~J-Z ( '-^pL4-r + c(eI -e Sp (c-9) Numerical results for - = - are tabulated in Tables 1C and 2C for case 1 and case 2, respectively, where g9 and 9 denote the temperature including axial conduction, Ql and 93 denote the temperature neglecting axial conduction, and Al and /~ denote the percent deviation of the temperature including axial conduction from the

TABLE 1C EFFECT OF AXIAL CONDUCTION FOR CASE 1 -- = O —L- - I ---- — ~ --- --,-;= — 0 --, e,* a, 9, A,| 9 A,i |, 0.5.414.437 -5.19.390.409 -4.76.346.356 -2.92 1.0.380.382 -.52.336.335.33.262.253 3.47 1.5.336.334.57.279.274 1.60.190.180 5.31 2.0.294.292.83.229.225 1.92.136.128 5.84 2.5.257.255.89.188.184 2.01.097.091 6.00 3.0.225.223.91.154.151 2.03.069.065 6.04 4.0.172.170.91.103.101 2.03.035.033 6.06 5.0.131.130.91.069.o68 2.03.018.017 6.o6 I \O I

TABLE 2C EFFECT OF AXIAL CONDUCTION FOR CASE 2 M =- 0.2-&7 I f M ^ 0<^?.~__e...L.... 0.5.855.874 -2.25.805.819 -1.62.717.712.70 1.0.765.764 0.17.678.670 1.11.529.507 4.51 1.5.673.668.74.559.549 1.80.380.361 5.61 2.0.589.584.87.458.449 1.97.272.257 5.93 2.5.515.510.90.375.368 2.02.194.183 6.02 3.0.450.446.91.307.301 2.03.138.130 6.05 4,0.344.341.91.206.202 2.03.070.o66 6.06 5.0.263.261.91.138.135 2.03.035.033 6.06 I 0\!

-61 - temperature neglecting axial conduction. The values $( = 0.269, 0.400, and 0O680 correspond to the wall temperature solution for N = 1, 2, and 20, respectively, when Fe=- i o A typical value of 1 =- was used in the calculations. Examination of the results shows that the effect of axial conduction in the end region due to end conditions decays rapidly and is small at z2 o The effect due to the imposed axial gradient on the surface.produces a constant percentage deviation from the solution neglecting axial conduction when _ 2 4. The magnitude of this deviation increases with A since the axial gradient increases with I o However, for significant stresses beyond - 7-. the value of / must be less than 1, approximatelyo Therefore, it is concluded that the effect of axial conduction is small for > 2- and may be neglectedo

APPENDIX D TABULATED VALUES OF THE EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR THE MODIFIED GRAETZ SOLUTION -62 -

-63 - TABLE 1D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR N = 20 1-il1 m 4 M (1 B 0 2.6069 1.4573.0494 0.0719 1 6.5099 -.7628 -.0632.0482 2 10.450 5382.0712.0383 3 14.403 -.4217 -.0768.0324 4 18 363.3489.0808.0282 5 22,328 -.2985 -.0840.0251 6 26.297.2612.0865.0226 7 30.269 -. 3223 -.0885.0206 8 34.244.2093.0901 o0189 9 38.220 -.1904 -o0915.0174 10 42.198.1747.0926 o0162 11 46.178 -.1613 -.0936.0151 12 50.159.1498.0943 o0141 13 541.42 -.1397 - 0950.0133 14 58.125.1309.0955.0125 15 62.109 -.1231 -.0959 o0118 16 66.094.1161 o0963.0112 17 70.080 -.1099 -.0966 o0106 18 74.67 o 1042.0968.0101 19 78.054 -.0991 -.0970 oo0096

-64 - TABLE 2D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR N = 17 AI6 A I _1) s_ 0 2.5905 1o4538 0578 o0840 1 6,4826 -.7549 - 0734.0554 2 10o415 5290 o08~2 o0435 3 14.363 - o4121 -.o881.0363 4 18,319 o3392 o0923.0313 5 22.281 - o2887 - 0955 0276 6 26.247 o2516 0979 0246 7 30.217 -.2229 - 0998 0222 8 34.189.2000.1012.0203 9 38.164 - 1814 -. 1024 o0186 10 42o141 o1658 o1033 o0171 11 46.120 -.1526 -o1040 o0159 12 50,100 o1413 o61046 o048 13 54 082 - o1315 -01050 o0138 14 58,065 1229 o1054.0129 15 62.049 - o1153 -.1056,0122 16 66,033 1085.1057 o0115 17 70o019 - o1024 - 1058 0108 18 74o006 o0970 o1059 0103 19 77~993 -.0920 - o1059 o 0097

-65 - TABLE 3D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR, N 6 L m / 114 a o, GR () s 1 ] 0 2.4072 1.4092.1530.2157 1 6.2036 -.6589 -.1767.1164 2 10.087.4262.1835.0782 3 14oo006 -.3112 -.1849.0575 4 17.945.2428 o1840.0447 5 21.897 -.1975 -.1822.0360 6 25.857.1655.1799.0298 7 29.824 -o1417 -.1774 o0251 8 33.796 1234.1749.0216 9 37.772 -.1089 -.1724 o0188 10 41.751.0972.1700 o0165 11 45.732 -.o876 -.1676 o0147 12 49.716.0795 1654 0132 13 53.701 -.0727 -.1632.0119 14 57.687 o0668 161 o0108 15 61.675 -.o0617 -1592 o0098 16 65.664.0573.1573.0090 17 69.653 -.0534 -.1554 o0083 18 73.644.0500 o1537 o0077 I 19 I 77 635 I. 0 469 I. -.1520 o0071

-66 - TABLE 4D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR N = 2. J I A41 J x _J C (I) 3_C 0 2.0000 1.2961.3679.4768 1 5.7439 -.4471 -.3328.1488 2 9.6451.2467.3027.0747 3 13.590 -.1635 -.2807.0459 4 17.555.1194.2640.0315 5 21.530 -.0926 -.2506 o.0232 6 25.511.0748.2396.0179 7 29.496 -.0623 -.2304.0143 8 33.484.0530.2224 o0118 9 37.474 - 0459 -.2155.0099 10 41.465 00403.2093.0084 11 45.458 -.0358 -.2039.0073 12 49.451.0321.1989 oo0064 13 53.446 -.0291 -.1944.0057 14 57.441.0265.1903.0050 15 61.437 -.0243 -.866.0045 16 65.433.0224.1831 o0041 17 69.429 -.0207 - 1799. 0037 18 73.426.0193.1769.0034 19 77.423 -.o18o 174o.o0031

-67 - TABLE 5D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR N = 1 iPA [, I 6 I) 6.1l 0 1.6413 1.2013.5497.6603 1 5 4783 -.2929 -.4079 o1195 2 9.4360.1467.3484 o0511 3 13415 -.0930 -.3133 0291 4 17.403 o0663.2892.0192 5 21.394 -o 50o6 -.2711 0137 6 25.388.0404.2569 o0104 7 29.383 -.0333 -.2453 o0082 8 33.379.0282.2355.0066 9 37.376 -.0243 o- 2272.0055 10 41.373.0212..2199 o0047 11 45.371.o0188 -.2135.oo40 12 49.369 o0168.2077.0035 13 53.367 -.0152 - 2026.0031 14 57.366 0138.1979 0027 15 61.364 -.0126 -.1937 0024 16 65.363 o0116.1897.0022 17 69.362 - 0107 -. 861.0020 18 73.361 o 0100.1828 o0018 19 77.360 -.0o093 -.1796.0017

-68 - TABLE 6D EIGENVALUES, EIGENFUNCTIONS AT THE WALL, AND COEFFICIENTS FOR N = 0o5 I j,CK [ (R ) _ _3e I 0 1.2716 1.1204 o7169 0.8032 1 5.2951 -.1710 -.4509.0771 2 9.3063 o0802.3725 o0299 3 13.312 -.0496 -.3299 o0164 4 17.316.0349 o3018.0105 5 21.318 -.0264 -.2813.0074 6 25.320.0210.2655.0056 7 29~321 -.0172 -2527.0044 8 33.322.0145.2420.0035 9 37.323 -.0125 -.2329.0029 10 41.324.0109.2251.0025 11 45.324 -.0096 -o2182 o0021 12 49.325 o0086 o2121 o018 13 53.325 -.0077 -.2066 o0016 14 57.326.0070.2017.004 15 61326 -.o0064 -.1971.0013 16 65.326.0059.1930.0011 17 69.327 -.0055 -.1892 o0010 18 73.327 o0051.1857 o0009 19 77327 - o0047 -.1824.0009 i

APPENDIX E THERMAL STRESS SOLUTION FOR PLANE STRAIN For the circular cylindrical geometry the plane strain analysis gives the stress distribution produced by radially distributed but axially and circumferentially uniform temperature distributions. However, it has been demonstrated that for "sufficiently smooth" temperature variations in the axial direction the one dimensional approximation will yield accurate resultso (See for instance, Boley and Weiner 3) The axisymmetric temperature expression is used in the plane strain solution and the axial dependence is carried as a parameter. Because the plane strain solution for thermal stresses in circular cylinders is well known and can be found in most text books (3) on thermal stresses, e.g. Boley and Weiner, the solution is given without further discussion as "r = iL R[ - i i- S C.R A ) RdR - i ~R Ri, Rda ] (D-1) RI PI X ^Rt =Jt i, p^ ( R,) + i (R,)Rd - ')Raj (D-2) When the end faces are free of external constraints the axial stress can be written in terms of CUr and A-t and is given by 0t = J(p * - (D-3) The shear stresses along coordinate directions are zero. The wall temperature distribution is represented by ^ Rt == ( I - 3 AI R ) Ae) (D-4) -69 -

where (i) is an arbitrary Equation (D-4) into Equation - I t(R"-1) - B ( Z-(PZ -70 -function of - Substitution of (D-l) and (D-2) yields B (1 Ro- R C-lR.]-) 3 R A R -- 4 CR 1] ) W (D-5) and <r; - t^s [ tJ [{ -[ -G - B( L RR - R)-G)} - {t [R -] - (LR/ R - '-1] )}-i-[i-BA;R When considering the exact temperature solution M b= and for the entrance region approximation (:) =l + * (-1) (cC)9 M-= (D-6) (D-7) (D-8)

BIBLIOGRAPHY 1. Arpaci, V. S. Conduction Heat Transfero Addison-Wesley, 1966. 2. Arpaci, V. S. Class Notes on "Thermoelasticity"o University of Michigan, 1965. 3. Boley, B. Ao, and Weiner, J. H. Theory of Thermal Stresses. John Wiley and Sons, 1960. 4. Boussinesq, J. "Application des Potentiels a 1'Etude de l'Equilibre et du Movement des Solides Elastiques." Gauthiers-Villars, Paris, 1885o 5. Galerkin, B. "Contribution a la Solution Generale du Probleme de la Theorie de l'Elasticite dans le cas de Trois Dimensions." Compto Rend., 190 (1930), 1047. 6. Goodier, J. N. "On the Integration of the Thermo-Elastic Equations." Phil. Mag., 23 (1937), 1017. 7. Jakob, M. Heat Transfer I. John Wiley and Sons, 1949. 8. Knudsen, J. G. and Katz, D. L. Fluid Dynamics and Heat Transfer. McGraw-Hill, 1958. 9. Kreith, F. Principles of Heat Transfer. 2nd Ed., International, 1965. 10. Love, A. E. H. Mathematical Theory of Elasticity. 4th Edo, Dover, (1944), 274. 11. Nowacki, W. Thermoelasticity. Addison-Wesley, 1960. 12. Papkovich, P. F. "Solution Generale des Equations Differentielles, Fondamentales d'Elasticite, Exprimee par Trois Functions Harmoniques" Compt. Rend., 195 (1932), 513. 13. Parkus, H. Instationare Warmespannungen. Springer-Verlag, 1959. 14. Parkus, H. "Thermal Stresses in Pipes." J. App. Mecho, 75 (1953), 485. 15. Schenk, J. and Dumore, J. M. "Heat Transfer in Laminar Flow Through Cylindrical Tubes." App. Sci. Res., 4A (1953), 39. 16. Schneider, P. J. "Effect of Axial Fluid Conduction on Heat Transfer in the Entrance Regions of Parallel Plates and Tubes." Heat Transfer and Fluid Mechanics Institute, (Preprints of papers) (1956), 41o -71 -

-72 -17. Van Wylen, G. J. and Sonntag, R. E. Fundamentals of Classical Thermodynamics. John Wiley and Sons, 1965. 18. Youngdahl, C. K. and Sternberg, E. "Transient Thermal Stresses in a Circular Cylinder." Jour. App. Mech., 84 (1961), 25.